Distinct equilateral triangle dissections of convex regions

Diane M. Donovan; James G. Lefevre; Thomas A. McCourt; Nicholas J. Cavenagh

Commentationes Mathematicae Universitatis Carolinae (2012)

  • Volume: 53, Issue: 2, page 189-210
  • ISSN: 0010-2628

Abstract

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We define a proper triangulation to be a dissection of an integer sided equilateral triangle into smaller, integer sided equilateral triangles such that no point is the vertex of more than three of the smaller triangles. In this paper we establish necessary and sufficient conditions for a proper triangulation of a convex region to exist. Moreover we establish precisely when at least two such equilateral triangle dissections exist. We also provide necessary and sufficient conditions for some convex regions with up to four sides to have either one, or at least two, proper triangulations when an internal triangle is specified.

How to cite

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Donovan, Diane M., et al. "Distinct equilateral triangle dissections of convex regions." Commentationes Mathematicae Universitatis Carolinae 53.2 (2012): 189-210. <http://eudml.org/doc/247011>.

@article{Donovan2012,
abstract = {We define a proper triangulation to be a dissection of an integer sided equilateral triangle into smaller, integer sided equilateral triangles such that no point is the vertex of more than three of the smaller triangles. In this paper we establish necessary and sufficient conditions for a proper triangulation of a convex region to exist. Moreover we establish precisely when at least two such equilateral triangle dissections exist. We also provide necessary and sufficient conditions for some convex regions with up to four sides to have either one, or at least two, proper triangulations when an internal triangle is specified.},
author = {Donovan, Diane M., Lefevre, James G., McCourt, Thomas A., Cavenagh, Nicholas J.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {equilateral triangle dissection; latin trade; equilateral triangle dissection; convex region; latin trade},
language = {eng},
number = {2},
pages = {189-210},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Distinct equilateral triangle dissections of convex regions},
url = {http://eudml.org/doc/247011},
volume = {53},
year = {2012},
}

TY - JOUR
AU - Donovan, Diane M.
AU - Lefevre, James G.
AU - McCourt, Thomas A.
AU - Cavenagh, Nicholas J.
TI - Distinct equilateral triangle dissections of convex regions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2012
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 53
IS - 2
SP - 189
EP - 210
AB - We define a proper triangulation to be a dissection of an integer sided equilateral triangle into smaller, integer sided equilateral triangles such that no point is the vertex of more than three of the smaller triangles. In this paper we establish necessary and sufficient conditions for a proper triangulation of a convex region to exist. Moreover we establish precisely when at least two such equilateral triangle dissections exist. We also provide necessary and sufficient conditions for some convex regions with up to four sides to have either one, or at least two, proper triangulations when an internal triangle is specified.
LA - eng
KW - equilateral triangle dissection; latin trade; equilateral triangle dissection; convex region; latin trade
UR - http://eudml.org/doc/247011
ER -

References

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  2. Cavenagh N.J., Donovan D.M., Khodkar A., Lefevre J.G., McCourt T.A., Identifying flaws in the security of critical sets in latin squares via triangulations, Australas. J. Combin. 52 (2012), 243–268. MR2917933
  3. Drápal A., On a planar construction of quasigroups, Czechoslovak Math. J. 41 (1991), no. 3, 538–548. MR1117806
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  5. Drápal A., Hämäläinen C., 10.1016/j.dam.2010.04.012, Discrete Applied Math. 158 (2010), no. 14, 1479–1495. Zbl1205.52014MR2659163DOI10.1016/j.dam.2010.04.012
  6. Drápal A., Hämäläinen C., Kala V., Latin bitrades, dissections of equilateral triangles and abelian groups, J. Combin. Des. 18 (2010), no. 1, 1–24. MR2584401
  7. Keedwell A.D., Critical sets in latin squares and related matters: an update, Util. Math. 65 (2004), 97–131. Zbl1053.05019MR2048415
  8. Laczkovich M., 10.1007/BF02122782, Combinatorica 10 (1990), no. 3, 281–306. Zbl0927.52028MR1092545DOI10.1007/BF02122782
  9. Laczkovich M., 10.1016/0012-365X(93)E0176-5, Discrete Math. 140 (1995), no. 1–3, 79–94. Zbl0822.05021MR1333711DOI10.1016/0012-365X(93)E0176-5
  10. Laczkovich M., Tilings of polygons with similar triangles, II, Discrete Comput. Geom. 19 (1998), no. 3, Special Issue, 411425, dedicated to the memory of Paul Erdös. Zbl0927.52028MR1608883
  11. McCourt T.A., On defining sets in latin squares and two intersection problems, one for latin squares and one for Steiner triple systems, PhD Thesis, University of Queensland, Australia, 2010. Zbl1195.05014MR2685159
  12. Tutte W.T., The dissection of equilateral triangles into equilateral triangles, Proc. Cambridge Philos. Soc. 44 (1948), 463–482. Zbl0030.40903MR0027521

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